3. A farm owner wants to fence off a rectangular pen against his barn, as shown. No fencing will be used against the barn. The two side pieces (each of length y) will cost $8/ft, and the front piece (of length x) will cost $10/ft. a) If a total of $1000 can be spent on the fencing, what should the dimensions of the pen be in order to maximize the area enclosed by the pen?
3. A farm owner wants to fence off a rectangular pen against his barn, as shown. No fencing will be used against the barn. The two side pieces (each of length y) will cost $8/ft, and the front piece (of length x) will cost $10/ft. a) If a total of $1000 can be spent on the fencing, what should the dimensions of the pen be in order to maximize the area enclosed by the pen?
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Transcribed Image Text:Title: Maximizing the Area of a Rectangular Pen
**Problem Description:**
A farm owner wants to construct a rectangular pen adjacent to his barn. No fencing is required along the barn side. The pen consists of two side fences and one front fence. The objective is to maximize the area enclosed by the pen with a specified budget for fencing.
**Cost Details:**
- Length of each side fence (y): Costs $8 per foot.
- Length of the front fence (x): Costs $10 per foot.
- Total budget for fencing: $1000.
**Diagram Explanation:**
The diagram illustrates the layout of the pen with the barn represented as a shaded area where no fencing is needed. The horizontal line (x) denotes the front fence, while the vertical lines (y) represent the side fences.
**Question:**
Given these conditions, what should be the dimensions of the pen to maximize the area within the allocated budget of $1000?
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